/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

reverse(g,a)

w.r.t. the given Prolog program could successfully be proven:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) PiDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) PiDP
(7) UsableRulesProof [EQUIVALENT, 0 ms]
(8) PiDP
(9) PiDPToQDPProof [SOUND, 0 ms]
(10) QDP
(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Clauses:

reverse(X1s, X2s) :- reverse(X1s, [], X2s).
reverse([], Xs, Xs).
reverse(.(X, X1s), X2s, Ys) :- reverse(X1s, .(X, X2s), Ys).


Query: reverse(g,a)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

reverse_in_2: (b,f)

reverse_in_3: (b,b,f)

Transforming Prolog into the following Term Rewriting System:

Pi-finite rewrite system:
The TRS R consists of the following rules:

   reverse_in_ga(X1s, X2s) -> U1_ga(X1s, X2s, reverse_in_gga(X1s, [], X2s))
   reverse_in_gga([], Xs, Xs) -> reverse_out_gga([], Xs, Xs)
   reverse_in_gga(.(X, X1s), X2s, Ys) -> U2_gga(X, X1s, X2s, Ys, reverse_in_gga(X1s, .(X, X2s), Ys))
   U2_gga(X, X1s, X2s, Ys, reverse_out_gga(X1s, .(X, X2s), Ys)) -> reverse_out_gga(.(X, X1s), X2s, Ys)
   U1_ga(X1s, X2s, reverse_out_gga(X1s, [], X2s)) -> reverse_out_ga(X1s, X2s)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2)  =  reverse_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

reverse_in_gga(x1, x2, x3)  =  reverse_in_gga(x1, x2)

[]  =  []

reverse_out_gga(x1, x2, x3)  =  reverse_out_gga(x1, x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)

reverse_out_ga(x1, x2)  =  reverse_out_ga(x1, x2)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   reverse_in_ga(X1s, X2s) -> U1_ga(X1s, X2s, reverse_in_gga(X1s, [], X2s))
   reverse_in_gga([], Xs, Xs) -> reverse_out_gga([], Xs, Xs)
   reverse_in_gga(.(X, X1s), X2s, Ys) -> U2_gga(X, X1s, X2s, Ys, reverse_in_gga(X1s, .(X, X2s), Ys))
   U2_gga(X, X1s, X2s, Ys, reverse_out_gga(X1s, .(X, X2s), Ys)) -> reverse_out_gga(.(X, X1s), X2s, Ys)
   U1_ga(X1s, X2s, reverse_out_gga(X1s, [], X2s)) -> reverse_out_ga(X1s, X2s)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2)  =  reverse_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

reverse_in_gga(x1, x2, x3)  =  reverse_in_gga(x1, x2)

[]  =  []

reverse_out_gga(x1, x2, x3)  =  reverse_out_gga(x1, x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)

reverse_out_ga(x1, x2)  =  reverse_out_ga(x1, x2)



----------------------------------------

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

   REVERSE_IN_GA(X1s, X2s) -> U1_GA(X1s, X2s, reverse_in_gga(X1s, [], X2s))
   REVERSE_IN_GA(X1s, X2s) -> REVERSE_IN_GGA(X1s, [], X2s)
   REVERSE_IN_GGA(.(X, X1s), X2s, Ys) -> U2_GGA(X, X1s, X2s, Ys, reverse_in_gga(X1s, .(X, X2s), Ys))
   REVERSE_IN_GGA(.(X, X1s), X2s, Ys) -> REVERSE_IN_GGA(X1s, .(X, X2s), Ys)

The TRS R consists of the following rules:

   reverse_in_ga(X1s, X2s) -> U1_ga(X1s, X2s, reverse_in_gga(X1s, [], X2s))
   reverse_in_gga([], Xs, Xs) -> reverse_out_gga([], Xs, Xs)
   reverse_in_gga(.(X, X1s), X2s, Ys) -> U2_gga(X, X1s, X2s, Ys, reverse_in_gga(X1s, .(X, X2s), Ys))
   U2_gga(X, X1s, X2s, Ys, reverse_out_gga(X1s, .(X, X2s), Ys)) -> reverse_out_gga(.(X, X1s), X2s, Ys)
   U1_ga(X1s, X2s, reverse_out_gga(X1s, [], X2s)) -> reverse_out_ga(X1s, X2s)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2)  =  reverse_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

reverse_in_gga(x1, x2, x3)  =  reverse_in_gga(x1, x2)

[]  =  []

reverse_out_gga(x1, x2, x3)  =  reverse_out_gga(x1, x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)

reverse_out_ga(x1, x2)  =  reverse_out_ga(x1, x2)

REVERSE_IN_GA(x1, x2)  =  REVERSE_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

REVERSE_IN_GGA(x1, x2, x3)  =  REVERSE_IN_GGA(x1, x2)

U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   REVERSE_IN_GA(X1s, X2s) -> U1_GA(X1s, X2s, reverse_in_gga(X1s, [], X2s))
   REVERSE_IN_GA(X1s, X2s) -> REVERSE_IN_GGA(X1s, [], X2s)
   REVERSE_IN_GGA(.(X, X1s), X2s, Ys) -> U2_GGA(X, X1s, X2s, Ys, reverse_in_gga(X1s, .(X, X2s), Ys))
   REVERSE_IN_GGA(.(X, X1s), X2s, Ys) -> REVERSE_IN_GGA(X1s, .(X, X2s), Ys)

The TRS R consists of the following rules:

   reverse_in_ga(X1s, X2s) -> U1_ga(X1s, X2s, reverse_in_gga(X1s, [], X2s))
   reverse_in_gga([], Xs, Xs) -> reverse_out_gga([], Xs, Xs)
   reverse_in_gga(.(X, X1s), X2s, Ys) -> U2_gga(X, X1s, X2s, Ys, reverse_in_gga(X1s, .(X, X2s), Ys))
   U2_gga(X, X1s, X2s, Ys, reverse_out_gga(X1s, .(X, X2s), Ys)) -> reverse_out_gga(.(X, X1s), X2s, Ys)
   U1_ga(X1s, X2s, reverse_out_gga(X1s, [], X2s)) -> reverse_out_ga(X1s, X2s)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2)  =  reverse_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

reverse_in_gga(x1, x2, x3)  =  reverse_in_gga(x1, x2)

[]  =  []

reverse_out_gga(x1, x2, x3)  =  reverse_out_gga(x1, x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)

reverse_out_ga(x1, x2)  =  reverse_out_ga(x1, x2)

REVERSE_IN_GA(x1, x2)  =  REVERSE_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

REVERSE_IN_GGA(x1, x2, x3)  =  REVERSE_IN_GGA(x1, x2)

U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)


We have to consider all (P,R,Pi)-chains
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(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
----------------------------------------

(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   REVERSE_IN_GGA(.(X, X1s), X2s, Ys) -> REVERSE_IN_GGA(X1s, .(X, X2s), Ys)

The TRS R consists of the following rules:

   reverse_in_ga(X1s, X2s) -> U1_ga(X1s, X2s, reverse_in_gga(X1s, [], X2s))
   reverse_in_gga([], Xs, Xs) -> reverse_out_gga([], Xs, Xs)
   reverse_in_gga(.(X, X1s), X2s, Ys) -> U2_gga(X, X1s, X2s, Ys, reverse_in_gga(X1s, .(X, X2s), Ys))
   U2_gga(X, X1s, X2s, Ys, reverse_out_gga(X1s, .(X, X2s), Ys)) -> reverse_out_gga(.(X, X1s), X2s, Ys)
   U1_ga(X1s, X2s, reverse_out_gga(X1s, [], X2s)) -> reverse_out_ga(X1s, X2s)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2)  =  reverse_in_ga(x1)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

reverse_in_gga(x1, x2, x3)  =  reverse_in_gga(x1, x2)

[]  =  []

reverse_out_gga(x1, x2, x3)  =  reverse_out_gga(x1, x2, x3)

.(x1, x2)  =  .(x1, x2)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)

reverse_out_ga(x1, x2)  =  reverse_out_ga(x1, x2)

REVERSE_IN_GGA(x1, x2, x3)  =  REVERSE_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(7) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(8)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   REVERSE_IN_GGA(.(X, X1s), X2s, Ys) -> REVERSE_IN_GGA(X1s, .(X, X2s), Ys)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

REVERSE_IN_GGA(x1, x2, x3)  =  REVERSE_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(9) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   REVERSE_IN_GGA(.(X, X1s), X2s) -> REVERSE_IN_GGA(X1s, .(X, X2s))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
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(11) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*REVERSE_IN_GGA(.(X, X1s), X2s) -> REVERSE_IN_GGA(X1s, .(X, X2s))
The graph contains the following edges 1 > 1


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(12)
YES